Have you read the novel Flatland? If you haven’t, maybe you should, since this book sets out the idea of the multidimensionality of the space and presents it in an unusual way.

More particularly, the protagonist of the book called “A Square” is a creature from a two-dimensional world, where everything is a plane, just like a sheet of paper. One day he is visited by another character, “A Sphere”, from a three-dimensional world (just like our own). A Square can’t imagine what the 3-D world looks like until A Sphere takes him from Flatland to Spaceland. From there A Square can see his world and family just like we see cartoons on the TV. The only problem is that two-dimensional creatures look disgusting because from Spaceland you can see their insides, since their skin has the shape of a square and does not cover their sides.

Then A Square wonders if a four-dimensional world is possible and what it would look like. In theory, it is, because there is no reason that could deny the possibility of the existence of space with as many dimensions as you can imagine. But no one can easily imagine, let’s say, a 10-dimensional world and the physical laws it would be ruled by.

But the question is: can the worlds with different dimensionality really interact with each other? The answer is no because they are ruled by different physical laws. In fact, A Square would have never been able to see his world from Spaceland since Flatland can interact only with forces of two-dimensional space. In order to see Flatland, the 3-D light from Spaceland should be able to interact with Flatland. But it is impossible since the objects in Flatland can’t reflect light or let it through. Remus Gogu in his book Book Riding. Creative Readings and Writings in Physics and Psychology writes, “If objects from Flatland could interact additionally in two distinct ways, then Flatland itself would become a 3-D world and its citizens would cease to exist because their organs would be exposed to outside forces (just as you would miss the skin on two sides of your body).”

However, the novel raises another interesting question: what a world with 4 and more dimensions would look like? Let’ begin with understanding how we perceive our three-dimensional world. In reality, we see in 2-D, just like when we watch a movie on the TV, except for the fact that we have a better perception of angles and distance due to the structure and position of our eyes.

“You cannot actually see the world in 3-D; when you look at a box in front of you, you may observe a side and a bit around the corners and edges, but you can never see all sides of the box at once”, writes Gogu. Thus, a 4-D creature would see a 4-D box as a complete 3-D image with some enhancement on edges and corners. Just imagine, wouldn’t it be amazing to see all the six sides of a box simultaneously with no cameras or mirrors? If you can’t imagine it, don’t worry; no one can! That’s why A Square couldn’t figure out what the 3-D world looks like until he visited Spaceland and saw it with his own eyes.

Anna LeMind

Anna is the founder and lead editor of the website Learning-mind.com. She is passionate about learning new things and reflecting on thought-provoking ideas. She writes about science, psychology and other related topics. She is particularly interested in topics regarding introversion, consciousness and subconscious, perception, human mind's potential, as well as the nature of reality and the universe.

One Comment

Thank you for bringing back to mind Edwin Abbot’s remarkable contribution, always a pleassure to contemplate. Surprised there’s not a dictionary entry for him in the form of a verb, “abbot —to foolishly dismiss, ignore, or fail to appreciate in the moment”.

For those who have read the book, the character referred to as Sphere passes through Flatland and causes quite a stir. Because the world of Flatland is but a plane, as Sphere passes though he manifests at first as a small circle, then as a progressively larger one, and then gets smaller and smaller again until he’s gone. Wow. And to make matters even worse, at any given stage of that progress Sphere’s form matched if not exceeded [in quality] that of Flatland’s priestly order This would have been highly offensive and not a little intimidating to those at the top of the ‘pierarchy’. There is however something of an unavoidable flaw in what defines Flatland. If we take a look at what we know about dimensionality that will come clear and, at the same time, will show that it may not reasonable [for us] to ever expect to see (experience) just ONE more dimension.

First, in addition to the latitude to anthropomorphize, let us grant of a 0th dimension, to wit motion itself, which is to say that even a completely barren landscape (contains no matter in any form) fundamentally allows for motion or even, for that matter, ensures of it. Just saying. Now add to that landscape a point. Just one point, infinitessimally small by definition. Ah, therefore no different in form, at least, than a completely barren landscape, and thus, if you’ll allow it, synonymous with the 0th dimension. So, a point, flitting about. But flitting about what? We would say, of course, flitting about space. But the point, being only a point, has no conception of that space.

Add now a second point anywhere within that same, let’s say, endless space. Instantly, by definition, we now have a line and, also by definition, the 1st dimension. But following again from the freedom/mandate to move, our line is ever stretching and ever coursing about and inevitably, at some point, will cross over itself. When it does so, a plane is formed. When this happens, the 2nd dimension is realized. And equally inevitably, by dint of this same process and the random formation of various planes at endlessly random locations, sooner or later that same line —and yet from an alternate path of approach— will cross itself again in our ‘neighborhood’ and, in so doing, will usher in a 3rd dimension, that is, anywhere plane crosses plane. Fun stuff. But is it really just a 3rd dimension? No. It is long understood to be four. Argument persists as to the nature of the 4th, which is largely an argument over how time should be defined or understood. But that there are four dimension, one of which subsumes of time, is fixed and known.

That’s where the unavoidable error comes through in Abbot’s story. He allows his characters to operate in time and, that is, with a consciousness of time. Not something a merely 2D world could accomplish. (Otherwise you’re allowing of time for all tiers which, by regression, allows for time to exist when nothing else does.) But nothing an appropriate degree of poetic license doesn’t allow for. This does however go to a larger point, meaning one of real substance. The shift from 2D to the 4D brought on merely by the intersection of the lesser with [essentially] itself, does not constitute a processional rate of change but rather an exponential one. The shift from two to four is either the doubling or the square of the lesser. This suggests that when and if 4D intersects 4D we get 8D (not 5D), as follows from either the cube of two or the doubling of four. So it’s not just one more layer of experience we’d obtain to but four new layers all at once. Again, fun stuff.

What might those layers be? Or, at least, of what sort? Honestly, the landscape itself seems pretty much locked down. This is to say that really the only variant seems to be that of the perceiver. Certainly that’s the crux of Abbot’s point. But even within Flatland there was, well, evidence of the next tier. The growing/shrinking temporary circles definitely registered as, let us say, an observable phenomenon. It just wasn’t well received. In this sense, there may be plenty of phenomena to which all or at least many of us are exposed which function as material evidence of the next tier and yet which for fear or dullness or the normal distractions of life we’ve not managed to explore or properly interpret. Furthermore, maybe that next tier isn’t so much quantitative in nature (x, y, z, and t) as it is qualitative in nature. Maybe it’s, say, an expansion of the four non-quantitative senses. (Arguably, with sight and sight alone we could have arrived at our current levels of understanding, while without sight we could not have.) Or maybe it’s quantitative after all, but only as the flipside of the four known dimensions, thereby giving us insight into legitimate forms of time travel, antigravity, and so forth. If so, maybe today’s crop circles are, at least by extension, an ironic tribute to Dr. Abbot.